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Properties

Label 2-185-185.103-c1-0-11
Degree $2$
Conductor $185$
Sign $-0.344 + 0.938i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.567 − 0.983i)2-s + (0.273 + 1.02i)3-s + (0.354 − 0.614i)4-s + (−1.79 − 1.33i)5-s + (0.849 − 0.849i)6-s + (−0.422 − 1.57i)7-s − 3.07·8-s + (1.63 − 0.941i)9-s + (−0.295 + 2.52i)10-s − 1.92i·11-s + (0.724 + 0.194i)12-s + (0.763 − 1.32i)13-s + (−1.31 + 1.31i)14-s + (0.873 − 2.19i)15-s + (1.03 + 1.79i)16-s + (1.14 − 0.660i)17-s + ⋯
L(s)  = 1  + (−0.401 − 0.695i)2-s + (0.157 + 0.589i)3-s + (0.177 − 0.307i)4-s + (−0.801 − 0.597i)5-s + (0.346 − 0.346i)6-s + (−0.159 − 0.596i)7-s − 1.08·8-s + (0.543 − 0.313i)9-s + (−0.0934 + 0.797i)10-s − 0.579i·11-s + (0.209 + 0.0560i)12-s + (0.211 − 0.366i)13-s + (−0.350 + 0.350i)14-s + (0.225 − 0.567i)15-s + (0.259 + 0.449i)16-s + (0.277 − 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507773 - 0.727213i\)
\(L(\frac12)\) \(\approx\) \(0.507773 - 0.727213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.79 + 1.33i)T \)
37 \( 1 + (-1.71 - 5.83i)T \)
good2 \( 1 + (0.567 + 0.983i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.273 - 1.02i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.422 + 1.57i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.92iT - 11T^{2} \)
13 \( 1 + (-0.763 + 1.32i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.14 + 0.660i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.17 + 0.583i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.57T + 23T^{2} \)
29 \( 1 + (-0.241 + 0.241i)T - 29iT^{2} \)
31 \( 1 + (1.00 + 1.00i)T + 31iT^{2} \)
41 \( 1 + (-6.76 - 3.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + (-1.79 + 1.79i)T - 47iT^{2} \)
53 \( 1 + (0.870 - 3.24i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.40 - 5.26i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.4 + 2.80i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.45 - 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.65 + 2.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.78 + 1.78i)T - 73iT^{2} \)
79 \( 1 + (12.3 - 3.30i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.29 + 8.57i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (16.4 + 4.40i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 15.4iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.05529334558805447330908272602, −11.22098124951682126062878651974, −10.29397825065435021396872594384, −9.541069581056046972946749767800, −8.549616105369676492938721317650, −7.31231342092150399900995499831, −5.82574120085966057092775459921, −4.31937196670702739489649245370, −3.23742641525247831617066519348, −0.952176022650350259815127629535, 2.46144139839006892216353934050, 3.99651990102053181674775130694, 5.94100851224898387509249897724, 7.06694511499966171723270186233, 7.60154487267998153674196114027, 8.514600774466965440693172187069, 9.723300572875545088473270593331, 11.07047352098185464432629369904, 12.18905697655480173527932315023, 12.56900097211587951362222230756

Graph of the $Z$-function along the critical line